MATH 6340: Commutative Algebra (Fall 2012)
Instructor: Ed Swartz
Textbook:
- Eisenbud: Commutative Algebra (recommended)
- Atiyah and MacDonald: Introduction to Commutative Algebra (recommended)
This course is an introduction to commutative algebra. Commutative algebra is a key ingredient of both algebraic number theory and algebraic geometry, and is a lively area of research on its own.
A typical list of topics includes:
- Groebner bases
- Ideals: localization, prime ideals, and operations on ideals
- Modules: graded rings and modules, syzygies, free resolution, tensor products, and operations on modules
- Noether normalization and Hilbert's nullstellensatz, and several other key concepts/theorems in commutative algebra: integral dependence, going up/down and integral closure.
- Primary decomposition in Noetherian rings
- Discrete valuation rings
- Hilbert functions and dimension theory
- Tor and Ext, and flatness
- Cohen-Macaulay rings, depth and regular sequences, Gorenstein rings, duality